We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we shall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another finer resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generalization of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like and how is it related to "classical" $BG$?

In other words as I explaned above from pure constructional point of view the classifying space $BG$ corresponds to the Cech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely its realization is the classifying space which we shall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Čech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Čech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Čech nerve provides a canonical resolution for calculation of Čech cohomology of $X$.

But we know that generally the resolution in Čech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one constructs another finer resolution using hypercovers of $X$. One can show that a resolution can be used to calculation of étale cohomology if it has certain coskeletal behavior in each stage. nevertheless, although the construction of such hypercovers is laborious, it is straightforward, so we have our "cooking recipe". So we can consider the hypercover as a natural, more powerful generalization of the Čech nerve.

My question is if we imitate literally the construction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Čech nerve, how the obtained simplicial object $\overline{BG}$ would look like and how is it related to "classical" $BG$?

In other words as I explained above from pure constructional point of view the classifying space $BG$ corresponds to the Čech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?

We can canonically via the bar construction associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we shall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another finer resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generalization of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like? How is it related to "classical" $BG$?

In other words as I explaned above from pure constructional point of view the classifying space $BG$ corresponds to the Cech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we shall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another finer resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generalization of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like and how is it related to "classical" $BG$?

In other words as I explaned above from pure constructional point of view the classifying space $BG$ corresponds to the Cech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?

We can canonically via the bar construction associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we sheall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another finer resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generalization of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like? How is it related to "classical" $BG$?

In other words as I explaned above from pure constructional point of view the classifying space $BG$ corresponds to the Cech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?

We can canonically via the bar construction associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps induced from two-sided bar construction $B(1,A,1)$ of $A$. The constructed object looks like

$$ N(\mathbf{B}A)= \left( \cdots A \times A \stackrel{\to}{\stackrel{\to}{\to}} A \stackrel{\to}{\to} {*} \right) $$

We know that if $A = G$ is a (finite, for sake of simplicity) group considered as category with one object and invertible morphisms, then $N(B A)$ or more precisely it's realization is the classifying space which we shall call $BG$.

On the other hand it is known that if $C$ is a category with pullbacks and $U \to X$ is a morphism. More generally we can consider instead of fix $U$ the covers $U = \coprod_i U_i $ of a covering sieve $\{U_i \to X \}$. The Cech nerve $C(U)$ is the simplicial object in $C$ that in degree $k$ is given by the $(k+1)$-fold fiber product of $U$ over $X$ with itself :

$$ C(U) = \left( \cdots U \times_X U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \times_X U \stackrel{\to}{\stackrel{\to}{\to}} U \right) $$

The way the Cech nerve and the bar are constructed is formally the same and this lead me to following question. It is well know (or more precisely that's exacly the purpose) that this Cech nerve provides a canonical resolution for calculation of Cech cohomology of $X$.

But we know that generally the resolution in Cech nerve not provides a resolution for calculation of eg etale cohomology. The reason is that is simply not "fine" enough. Instead one consructs another finer resoluton using hypercovers of $X$. One can show that a resolution can be used to calculation of etale cohomology if it has certain coskeletal bahavior in each stage. nevertheless, although the construction of such hypercover is laborious, it is straigh forward, so we have our "cooking recipe". So we can consider the hypercover as natural, more powerful generalization of Cech nerve.

My question is if we imitate literally the contruction of a hypercovering of scheme or stack $X$ on (let say finite) group $A:=G$ as above in same manner as we did for bar construction and Cech nerve, how the obtained simplicial object $\overline{BG}$ would look like? How is it related to "classical" $BG$?

In other words as I explaned above from pure constructional point of view the classifying space $BG$ corresponds to the Cech nerve $C(U)$ of the cover $U \to X$. And I'm asking if there exist a simplicial object (or respectively it's realization as top space) $\overline{BG}$ which corresponds in analogous manner to a resolution by hypercover (over a scheme/stack)? Does this $\overline{BG}$ have some nice properties, for example does it represent some interesting objects as in case of classical $BG$?